Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-05-10T17:25:48.246Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous Hahn Polynomials of Differential Operator Argument and Analysis on Riemannian Symmetric Spaces of Constant Curvature

Published online by Cambridge University Press:  20 November 2018

Erich Badertscher  and
Tom H. Koornwinder
Erich Badertscher
Affiliation:
Mathematisches Institut, Sidlerstrasse 5, CH-3012 Bern, Switzerland email: badertscher@mai.unibe.ch
Tom H. Koornwinder
Affiliation:
CWI, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For the three types of simply connected Riemannian spaces of constant curvature it is shown that the associated spherical functions can be obtained from the corresponding (zonal) spherical functions by application of a differential operator of the form p(i d/dt), where p belongs to a system of orthogonal polynomials: Gegenbauer polynomials, Hahn polynomials or continuous symmetric Hahn polynomials. We give a group theoretic explanation of this phenomenon and relate the properties of the polynomials p to the properties of the corresponding representation. The method is extended to the case of intertwining functions.

Keywords

43A9043A8533C8033C45spaces of constant curvaturespherical functionsassociated spherical functionsBessel functionsJacobi functionsGegenbauer polynomialsHahn polynomialscontinuous symmetric Hahn polynomials.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 4 , 01 August 1992 , pp. 750 - 773
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Askey, R. and Wilson, J., A set of hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 13 (1982), 651–655.Google Scholar
2. Askey, R. and Wilson, J., Some basic hyper geometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. (319) 54 (1985).Google Scholar
3. Chihara, T.S., An introduction to orthogonal polynomials, Gordon and Breach, 1978.Google Scholar
4. Dixmier, J. and Malliavin, P., Factorisations de fonctions et de vecteurs indéfiniment differentiables, Bull. Sci. Math. (2) 102 (1978), 307–330.Google Scholar
5. Erdélyi, A., Zur Théorie der Kugelwellen, Physica 4 (1937), 107–120.Google Scholar
6. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G., Higher transcendental functions, Vol. I, McGraw-Hill, 1953.Google Scholar
7. Erdélyi, A., Magnus, W., Higher transcendental functions, Vol. II, McGraw-Hill, 1953.Google Scholar
8. Faraut, J., Distributions sphériques sur les espaces hyperboliques, J. Math. Pures Appl. (9) 58 (1979), 369- 444.Google Scholar
9. Faraut, J., Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques, in Analyse harmonique, CIMPA, Nice, 1982.315–446.Google Scholar
10. Faraut, J., Analyse harmonique et fonctions spéciales, in Deux cours d'analyse harmonique, Birkhaiiser, Boston, 1987.1–151.Google Scholar
11. Flensted-Jensen, M., Non-Riemannian symmetric spaces, CBMS Regional Conference Series 61, American Mathematical Society, 1986.Google Scholar
12. Gelfand, I.M. and Shilov, G.E., Generalized functions, Properties I. and operations, Academic Press, 1964.Google Scholar
13. Helgason, S., A duality for symmetric spaces with applications to group representations, II. Differential equations and eigenspace representations, Adv. in Math. 22 (1976), 187–219.Google Scholar
14. Helgason, S., Groups and geometric analysis, Academic Press, 1984.Google Scholar
15. Karlin, S. and McGregor, J.L., The Hahnpolynomials, formulas and an application, Scripta Math. 26( 1961 ), 33–46.Google Scholar
16. Koornwinder, T.H., The representation theory o/SL(2, R), a global approach, Report ZW 145/80 , Mathematisch Centrum, Amsterdam, 1980.Google Scholar
17. Koornwinder, T.H., Jacobi functions and analysis on noncompact semisimple Lie groups, In: Special functions: Group theoretical aspects and applications (ed. Askey, R.A., Koornwinder, T.H. and Schempp, W.), Reidel, 1984. 1–85.Google Scholar
18. Koornwinder, T.H., Group theoretic interpretations of Askey's scheme of’ hyper geometric orthogonal polynomials, in Orthogonal polynomials and their applications, (ed. Alfaro, M., Dehesa, J.S., Marcellan, F.J., Rubio, J.L. de Francia and Vinuesa, J.), Lecture Notes in Math. 1329, Springer, 1988.46–72.Google Scholar
19. Lirnic, N., Niederle, J. and Raczka, R., Continuous degenerate representations of noncompact rotation groups. II, J. Math. Phys. 7 (1966), 2026–2035.Google Scholar
20. van, B. der Pol, A generalization of Maxwell's definition of solid harmonics to waves in n dimensions, Physica 3 (1936), 393–397.Google Scholar
21. Strasburger, A., Inducing spherical representations of semi-simple Lie groups, Dissertationes Math. (Rozprawy Mat.) 122 (1975).Google Scholar
22. Szegö, G., Orthogonal polynomials, Amer. Math. Soc. Colloquium Publications 23, American Mathematical Society, Fourth edition, 1975.Google Scholar
23. Ya, N.. Vilenkin, Special functions and the theory of group representations, Translations of Mathematical Monographs 22, American Mathematical Society, 1968.Google Scholar
24. Wallach, N.R., Real reductive groups I, Academic Press, 1988.Google Scholar
25. Watson, G.N., A treatise on the theory of Bes seI functions, Cambridge University Press, Second edition, 1944.Google Scholar